A275699 Excess of numbers that are not squarefree.

1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 1, 3, 1, 1, 3, 3, 1, 2, 1, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 2, 1, 1, 2, 1, 6, 1, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 5, 2, 1, 3, 1, 1, 3, 1, 4, 1, 4, 2, 1

Offset

1,2

Comments

The "excess" of a number is the number of prime divisors with multiplicity (the Omega function, A001222) minus the number of distinct prime divisors (the omega function, A001221). A046660(n) gives the excess of n.

Since squarefree numbers have no excess, this sequence is essentially A046660 with the 0's removed.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = A046660(A013929(n)).

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p-1)) / (1-6/Pi^2) = A136141/A229099 = 1.9719717... - Amiram Eldar, Feb 10 2021

Example

Since 16 = 2^4, 16 has four prime divisors, but only one distinct divisor. Hence Omega(16) - omega(16) = 4 - 1 = 3. As 16 is the fifth number that is not squarefree, its corresponding 3 is a(5) in this sequence.
17 is prime and thus has no excess and no corresponding term in this sequence.
18 = 2 * 3^2, Omega(18) - omega(18) = 3 - 2 = 1, thus a(6) = 1.

Mathematica

DeleteCases[Table[PrimeOmega[n] - PrimeNu[n], {n, 200}], 0] (* Alonso del Arte, Aug 05 2016 *)

Prog

(PARI) for(n=1, 200, if(bigomega(n)!=omega(n), print1(bigomega(n)-omega(n), ", ")))

Crossrefs

Cf. A001221, A001222, A013929, A046660, A136141, A229099.

Sequence in context: A124032 A254046 A137457 * A305633 A214123 A369188

Adjacent sequences: A275696 A275697 A275698 * A275700 A275701 A275702

Keyword

nonn

Author

Felix Fröhlich, Aug 05 2016

Status

approved